Fθ lens and image apparatus using the same

ABSTRACT

An fθ lens acting to effect uniform speed scanning while causing a light beam deflected at a uniform angular velocity by the deflecting surface of a deflector to be imaged on a plane to be scanned, wherein the shape of the curved surface of at least the lens surface of the lens which is adjacent to the deflection point side of the deflecting surface in the main scanning plane is an aspherical shape and particularly, near the optical axis, in at least the main scanning plane, this aspherical shape is convex toward the deflection point, and when the radius of curvature of this convex shape near the optical axis in the main scanning plane is r 1  and the focal length of the fθ lens near the optical axis in the main scanning plane is fm, 
     
         0≦r.sub.1 &lt;|fm| 
    
     and yet, when with the point of intersection between the lens surface adjacent to the deflection point side and the optical axis as the origin and with the coordinates system of the x-axis plotted in the direction of the optical axis and the coordinates system of the y-axis plotted in the main scanning plane perpendicularly thereto, the surface shape in the main scanning plane is expressed as a function S 1  (y) in which y is a variable, and when the maximum effective diameter of this surface in the main scanning plane is Ymax, S 1  (y) is defined between 0 and Ymax, and when r 1  &lt;Ymax, 
     
         -1&lt;S.sub.1 (r.sub.1)/r.sub.1 &lt;0.5 
    
     and when r 1  ≧Ymax, 
     
         -1×Ymax/r.sub.1 &lt;S.sub.1 (Ymax)/Ymax&lt;0.5×Ymax/r.sub.1.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to an fθ lens used in a laser scanning opticalsystem or the like, and particularly to an fθ lens comprised of a lensand an image forming apparatus using the same.

2. Related Background Art

Heretofore, in a scanning optical system, chiefly a laser source such asa semiconductor laser has been used as a light source and light from thelight source has been imaged on a plane to be scanned through an opticalsystem to thereby form a laser spot, and the laser source has beensubjected to modulation conforming to image information while the laserspot is scanned on the plane to be scanned by a deflector such as apolygon mirror disposed in the optical path, whereby a desired image hasbeen formed on the plane to be scanned.

Where such a laser scanning optical system is particularly an opticalsystem which scans a laser by the rotation of a deflector such as apolygon mirror, it is desirable that a laser spot be uniformly formed onthe whole plane to be scanned (that is, the curvature of image field ofa lens system be corrected) and the angle of rotation of the deflectorand the position of the spot formed on the plane to be scanned have aproportional relation (that is, the speed of the spot scanned on theplane to be scanned become constant on the whole plane to be scanned,i.e., have constant speed property.). For this purpose, a correctingoptical system is used between the polygon mirror and the plane to bescanned. Usually, this lens for correction used in the laser scanningsystem is a lens called an fθ lens having such a distortioncharacteristic that the angle of incident light and the image height arein a proportional relation, so as to have constant speed property.

Further, even where a mirror surface which is the deflecting surface ofthe deflector has an inclination, it is regarded as desirable from theviewpoints of the manufacture of the deflector and the quality of imageto have optical correcting means so that the position of the scanningline may not deviate (inclination correcting function).

There have been made numerous inventions relating to an fθ lens havingsuch inclination correcting function. Particularly, regarding a type inwhich the number of constituent lenses is two or more, many inventionshave been made and put into practical use.

In contrast, several fθ lenses in which the lens system is constructedof a lens have also been devised as simpler optical systems.

As seen, for example, in Japanese Laid-Open Patent Application Nos.55-7727 and 58-5706, there is an example in which an fθ lens iscomprised of a spherical lens.

Also, like Japanese Laid-Open Patent Application Nos. 63-50812 and1-224721, there is an example in which a toric surface is used as a lenssurface and yet an fθ lens is constructed.

Further, in Japanese Laid-Open Patent Application Nos. 54-87540 and54-98627, there is disclosed an example in which an aspherical surfaceterm is introduced into a single lens, and in Japanese Laid-Open PatentApplication Nos. 62-138823, 63-157122 and 2-87109, there is disclosed anexample in which a high-order aspherical surface is introduced into alens surface to thereby construct an fθ lens.

However, in the plano-convex fθ lens disclosed in Japanese Laid-OpenPatent Application No. 55-7727 among the above-mentioned examples of theprior art, constant speed property is corrected, but curvature of imagefield remains and it is difficult to keep the uniformity of spot. Inthis case, if design is made such that the size of the effectivescanning width by this lens is sufficiently large relative to the planeto be scanned, curvature of image field can be made small, but theentire apparatus will become bulky, and this cannot be said to bepractical.

In Japanese Laid-Open Patent Application No. 58-5706, the fθ lenscomprising a spherical lens is made into a meniscus lens having itsconcave surface facing the deflecting surface side to thereby accomplishaberration correction, but as in Japanese Laid-Open Patent ApplicationNo. 55-7727, an attempt to sufficiently correct curvature of image fieldand constant speed property at a time would give rise to the necessityof making the distance between this lens and the plane to be scannedgreat.

In Japanese Laid-Open Patent Application No. 63-50812, a toric lens isused to make a lens in which curvature of image field and constant speedproperty are corrected. Particularly, the aberrations in the mainscanning plane (curvature of image field and constant speed property)and the aberrations in the sub-scanning plane perpendicular to the mainscanning plane can be corrected independently and therefore, bettercorrection than in the aforedescribed two examples is accomplished.Further, the deflecting surface and the plane to be scanned are madeinto a conjugate relation on the sub-scanning cross-section, therebyaccomplishing the correction of the inclination of the mirror surface ofthe deflector as well. The distance from the deflecting surface to theplane to be scanned is also kept relatively small. However, a toriclens, if viewed only in the main scanning plane, is equivalent to aspherical single lens, and it is difficult to correct curvature of imagefield and constant speed property at a time. Therefore, to provide alens shape in which aberrations are good, it is necessary to make thethickness t of the lens considerably great relative to the focal lengthf, i.e., 0.3<t/f<0.5, as described in this patent application. For thisreason, as an actual lens, the manufacture thereof is difficult and theuse of a single lens will lead to no merit.

In Japanese Laid-Open Patent Application No. 1-224721, a toric lens isalso used and further, design is made such that convergent light iscaused to enter the toric lens, thereby accomplishing aberrationcorrection, but as in Japanese Laid-Open Patent Application No.63-50812, even in the main scanning plane alone, it is difficult to makecurvature of image field and constant speed property compatible. In thisexample, as described in the specification thereof, constant speedproperty is restricted to such a degree of correction that it can beelectrically corrected (expressed as substantially constant speed in thespecification), whereby the correction of curvature of image field iseffected preponderantly. Thus, during image writing, the timing of imageinformation is varied to thereby correct the distortion of the writtenimage. In this case, however, constant speed property is under-correctedand thus, the speed of the spot on the plane to be scanned varies at alltimes, and the quantity of light received per unit time by the plane tobe scanned is varied. It is possible to correct this by varying thequantity of laser light produced from the light source, but this willlead to too many correcting circuits, which does not mean the obtainmentof the merit of using a single lens.

In the case of a single lens whose lens shape is a spherical shape inthe main scanning plane as in these examples, it has been difficult torealize an fθ lens having a sufficient characteristic.

In contrast, as described in Japanese Laid-Open Patent Application Nos.54-87540 and 54-98627, there is known an example in which an asphericalterm is introduced to construct an fθ lens.

The aspherical lenses described in Japanese Laid-Open Patent ApplicationNos. 62-138823, 63-157122 and 1-99013 are such that the shape of thelens near the optical axis thereof is that of a meniscus lens in whichthe deflecting surface side is a concave surface or a biconvex lens, andin any of these lenses, the thickness t of the lens is great relative tothe width of the plane to be scanned.

The example shown in Japanese Laid-Open Patent Application No. 62-138823is a meniscus lens in which the deflecting surface side is a concavesurface or a biconvex lens, but the thickness of the lens is 20 mm atsmallest relative to the effective scanning width of 200 mm.

The example shown in Japanese Laid-Open Patent Application No. 63-157122is a meniscus lens in which near the optical axis, the deflectingsurface side is a concave surface, but the thickness of the lens is 30mm at smallest relative to the maximum scanning width (the scanningwidth corresponding to the angle of view mentioned as the maximumscanning angle θmax in the specification; 254 mm or so in anyembodiment).

The example shown in Japanese Laid-Open Patent Application No. 2-87109is a lens which is biconvex near the optical axis and therefore, thethickness of the lens is also great.

It is difficult to make such aspherical lenses by carrying out theprocessing as is done in the manufacture of ordinary spherical lenses.Therefore, processing and molding are carried out by the use ofmaterials such as plastics rich in workability to make such asphericallenses, thereby solving the problem in manufacture. However, plasticsare generally apt to be affected by environmental fluctuations, and areparticularly apt to be varied in refractive index by humidity andtemperature. Especially, where the thickness of the lens is great, alight beam passing therethrough is greatly subjected to a variation inrefractive index and thus, the imaging position is varied byenvironmental fluctuations. Further, a great thickness of the lens is afactor which increases internal homogeneity, distortion, moldingtermination time, etc. in carrying out processing and molding.

Thus, the aspherical lenses known as the above-described three examplesof the prior art could not be said to be practical because of theirgreat lens thicknesses. Particularly in these examples, the shape of thedeflection point side lens surface in the main scanning plane is notappropriate and therefore, to satisfy the performance necessary as an fθlens, it is necessary to make the lens thicker than the practical range,and this has been low in realizability.

SUMMARY OF THE INVENTION

The present invention has been made in view of the above-notedcircumstances and the object thereof is to provide an fθ lens whichsatisfies the performance required of an fθ lens and yet whose lensthickness is kept small and which is suitable for plasticization andcomprises a single lens, and an image forming apparatus which uses suchfθ lens, whereby the apparatus can be made compact and thin.

The above object of the present invention is achieved by the fθ lens ofthe present invention which will hereinafter be described and an imageforming apparatus using the same.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional view of a first embodiment of the presentinvention in the main scanning plane.

FIGS. 2A and 2B illustrate the shape of the main scanning cross-sectionof the surface shape of the fθ lens of the present invention which isadjacent to the deflection point side, FIG. 2A referring to a case wherer₁ <Ymax, and FIG. 2B referring to a case where r₁ ≧Ymax.

FIG. 3 illustrates the arrangement of the entire scanning system of thefθ lens of the present invention.

FIG. 4 illustrates the state of a light beam incident on the fθ lens ofthe present invention in the main scanning plane.

FIG. 5 illustrates the radius of curvature of the fθ lens of the presentinvention in the sub-scanning plane.

FIG. 6 is a cross-sectional view of a second embodiment of the presentinvention in the main scanning plane.

FIG. 7 is a cross-sectional view of a third embodiment of the presentinvention in the main scanning plane.

FIG. 8 is a cross-sectional view of a fourth embodiment of the presentinvention in the main scanning plane.

FIG. 9 is a cross-sectional view of a fifth embodiment of the presentinvention in the main scanning plane.

FIG. 10 is a cross-sectional view of a sixth embodiment of the presentinvention in the main scanning plane.

FIG. 11 is a cross-sectional view of a seventh embodiment of the presentinvention in the main scanning plane.

FIG. 12 is a cross-sectional view of an eighth embodiment of the presentinvention in the main scanning plane.

FIGS. 13 to 20 show aberrations in the embodiments of the fθ lens of thepresent invention.

FIG. 21 is a cross-sectional view of an image forming apparatus usingthe fθ lens of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the fθ lens of the present invention which is an fθ lens comprised ofa lens, the shape of the curved surface of at least the lens surface ofsaid lens which is adjacent to the deflection point side of a deflectorin the main scanning plane is an aspherical shape and particularly, nearthe optical axis, in at least the main scanning plane, this asphericalshape is convex toward the deflection point, and when the radius ofcurvature of this convex shape near the optical axis in the mainscanning plane is r₁ and the focal length of the fθ lens near theoptical axis in the main scanning plane is fm, there is the relationthat

    0≧r.sub.1 <|fm|                   (1)

between r₁ and the absolute value of fm and yet, when with the point ofintersection between the lens surface adjacent to the deflection pointside and the optical axis as the origin and with the coordinates systemof the x-axis plotted in the direction of the optical axis and thecoordinates system of the y-axis plotted in the main scanning planeperpendicularly thereto, the surface shape in the main scanning plane isexpressed as a function S₁ (y) in which y is a variable, and when themaximum effective diameter of this surface in the main scanning plane isYmax and S₁ (y) is defined between 0 and Ymax, and when as shown, therelation between the aspherical shape and the radius of curvature nearthe optical axis is r₁ <Ymax, that is, when a spherical lens having theradius of curvature r₁ cannot have the effective diameter Ymax (e.g.FIG. 2-a), the lens shape S₁ (r₁) at a position y=r₁ satisfies

    -1<S.sub.1 (r.sub.1)/r.sub.1 <0.5                          (2a)

and when r₁ >Ymax, that is, when a spherical lens having the radius ofcurvature r₁ can have the effective diameter Ymax (e.g. FIG. 2-b), thelens shape S₁ (Ymax) at a position y=Ymax satisfies.

    -1×Ymax/r.sub.1 <S.sub.1 (Ymax)/Ymax

    <0.5×Ymax/r.sub.1                                    (2b)

whereby a more practical fθ lens is realized. In this case, if the shapeof the lens surface adjacent to the deflection point side in the mainscanning plane is convex toward the deflection point and does notsatisfy expression (1), the constant speed property near the opticalaxis will be under-corrected, and if said shape does not satisfyexpression (2a) or expression (2b), the correction of the curvature ofimage field in the main scanning surface will become insufficient. Thatis, it is the condition necessary for the fθ lens of the presentinvention to satisfy the performance required of an fθ lens thatexpression (1) and expression (2a) or (2b) be satisfied at a time.

Further, in the fθ lens of the present invention, even when a materialwhose refractive index is varied by an environmental fluctuation such asthe fluctuation of humidity or temperature is chosen as the material ofthe lens, so that the influence thereof may be small, the thickness t ofthe lens, relative to the effective scanning width L (see FIG. 3), canbe kept in a range expressed as

    0<t/L<0.08                                                 (3)

or when the distance from the deflecting surface of the lens to theplane to be scanned is g, the thickness t can be kept in a rangeexpressed as

    0<t/g<0.15                                                 (4)

and the fluctuation of the imaging position by an environmentalfluctuation can be kept small.

Further, the choosing the lens shape so that the shape of the lens nearthe optical axis may be a convex meniscus lens in at least the mainscanning plane, the thickness of the lens can be kept small and yet, thethickness of the lens near the optical axis and the thickness of thelens in the effective end portion thereby can be made not to differgreatly from each other to thereby improve the homogeneity duringprocessing and molding, and processing conditions such as distortion andprocessing time.

Also, by choosing the focal length fm of the lens near the optic axis inthe main scanning plane relative to the distance g from the deflectingsurface to the plane to be scanned so as to satisfy

    |fm|/g>0.8                               (5)

and by using an aspherical lens, assemblage property such as themounting accuracy of the lens can be prevented from being remarkablyaggravated. In this case, the sign of fm may be +or-.

Also, if the light beam incident on this lens is made into convergentlight in at least the main scanning plane, the distance from the lens tothe plane to be scanned can be shortened to thereby make the size of theentire scanning system small. In this case, when the distance from thedeflection point to a point at which this convergent light is imaged inthe main scanning plane is p and the distance from the deflection pointto the plane to be scanned is g (see FIG. 4), design is made such thatconvergent light which will satisfy the condition that

    0.7<p/g                                                    (6)

is caused to be incident on the lens, the aggravation of the constantspeed property in the end portion of the plane to be scanned which iscaused by the lens being made compact can be suppressed within apractical range. On the other hand, if this incident light beam is madeinto substantially parallel light in at least the main scanning plane,there will be a limit in making the lens compact, but it will becomepossible to alleviate the mutual mounting accuracy of the deflectionpoint and the fθ lens.

In the fθ lens of the present invention, the lens shape is not made intoan axis-symmetrical lens shape and with regard to at least one of thecurved surface on the deflection point side and the curved surface onthe side of the plane to be scanned, the radius of curvature near theoptical axis differ between the main scanning cross-section and thesub-scanning cross-section (a cross-section perpendicular to the mainscanning cross-section and containing the optical axis), whereby thecurvature of image field in the sub-scanning direction (a directionperpendicular to the direction in which the light beam is scanned by anoptical deflector and the direction of the optical axis) can be keptsmall than when the lens is constructed of only an axis-symmetricalshape.

Figure, with regard to at least one of the curved surface on thedeflection point side and the curved surface on the side of the plane tobe scanned, design is made such that the radius of curvature r' (seeFIG. 5) of the curved surface of the lens determined in a planecontaining a normal in the main scanning plane and perpendicular to themain scanning plane continuously varies in the effective portion of thelens, whereby the curvature of image field in the sub-scanning directioncan be better corrected.

In this case, if in the sub-scanning cross-section about the opticalaxis, the deflection point and the plane to be scanned are designed soas to be in an optically conjugate relation, the inclination of thedeflecting surface of the deflector can be corrected in advance.

FIG. 1 shows a first embodiment of the fθ lens according to the presentinvention. FIG. 1 is a cross-sectional view of the lens as it is seen inthe main scanning plane. The design values of this lens are shown inTable 1, and the performance of this lens is shown in Table 2. The mainscanning plane refers to a light beam surface the light beam deflectedby the deflecting surface of the deflector forms with the lapse of time.

The cross-sectional shape of this lens in the main scanning plane, whenwith the point of intersection between the lens surface and the opticalaxis as the origin, the coordinates system of the x-axis is plotted inthe direction of the optic axis and the coordinates system of the y-axisis plotted in the main scanning plane perpendicularly to the directionof the optical axis, is developed into the form of a polynominalincluding the high-order term that ##EQU1## where r is the radius ofcurvature of the lens surface in the main scanning plane near theoptical axis. In this case, with regard to respective surfaces, as shownin Table 1, the cross-sectional shape of the lens surface adjacent tothe deflection point of the deflecting surface is expressed as afunction S₁ (y) in which y is a variable, the radius of curvature in themain scanning plane near the optical axis is expressed as r₁, thecoefficients of the high order term are expressed as k₁, B₁, C₁, D₁ andE₁, the cross-sectional shape of the lens surface adjacent to the planeto be scanned is expressed as a function S₂ (y) in which y is avariable, the radius of curvature in the main scanning plane near theoptical axis is expressed as r₂, and the coefficients of the high-orderterm are expressed as k₂, B₂, C₂, D₂ and E₂. With regard to the lenssurface adjacent to the deflection point, the functions S₁ (y) and r₁are shown in FIG. 2 with the maximum effective diameter in the mainscanning plane as Ymax.

With regard to the arrangement of the entire scanning system, as shownin FIG. 3, the distance from the deflection point of the opticaldeflector to the entrance surface of the lens is a, the thickness of thelens on the optical axis is t, the distance from the exist surface ofthe lens to the plane to be scanned is b, the distance from thedeflection point to the plane to be scanned is g, the maximum scanningangle of the lens is θmax, and the effective scanning width on the planeto be scanned is L. FIG. 3 shows a cross-sectional view of the scanningsystem as it is seen in the main scanning plane.

Also, the incident light onto the optical deflector, as shown in FIG. 4,is convergent light which is imaged at a distance p from the deflectionpoint, in the main scanning plane. In the sub-scanning plane, the lightbeam emitted from a light source, not shown, passes through acylindrical lens, not shown, and enters so as to be imaged on thedeflection point of the optical deflector. In the sub-scanning plane,the lens assumes the construction of a so-called inclination correctingoptical system.

The shape of this lens in the sub-scanning direction, when the radius ofcurvature of the lens surface near the optical axis in the sub-scanningcross-section containing the optical axis is r_(s) and as shown in FIG.5, the z-axis is plotted in a direction containing a normal to thecurved surface of the lens in the main scanning plane and perpendicularto the main scanning plane and the radius of curvature determined in aplane containing the normal and the z-axis is r' and when the positionof the foot of this normal is at a position of y from the optical axis,is expressed by the use of a variable which gives a variation in radiusof curvature in the form of ##EQU2## where r'=r_(s) (1+Ay² +By⁴ +Cy⁶+Dy⁸ +Ey¹⁰). Thus, the shape of the lens in the sub-scanning directionperpendicular to the main scanning plane varies continuously inaccordance with the value of y. In this case, with regard to eachsurface, as shown in Table 1, the lens shape of the lens surfaceadjacent to the deflection point in the sub-scanning plane is expressedas S'₁ (y), the radius of curvature in the sub-scanning plane near theoptical axis is expressed as r₁, coefficients which give a variation incurvature are expressed A'₁, B'₁, C'₁ and E'₁, the lens shape of thelens surface adjacent to the plane to be scanned in the sub-scanningplane is expressed as S'₂ (y), the radius of curvature in thesub-scanning plane near the optical axis is expressed as r₂, andcoefficients which give a variation in curvature are expressed as A'₂,B'₂, C'₂, D' ₂ and E'₂.

In this lens, with regard to the sub-scanning cross-section of the lens,the deflection point and the sub-scanning cross-section are in aconjugate relation, thereby correcting inclination.

In the case of the first embodiment, S'₁ (y) is constant over the wholelens surface and r'=∞.

FIG. 6 shows a second embodiment of the fθ lens according to the presentinvention. FIG. 6 shows a cross-sectional view of the lens as it is seenin the main scanning plane. The design values of this lens are shown inTable 3, and the performance of this lens is shown in Table 4.

In the second embodiment, the shape of the lens in the main scanningcross-section is the same as that in the first embodiment, but the shapeof the lens in the sub-scanning direction differs from that in the firstembodiment. Particularly, the surface adjacent to the deflector is anaspherical surface symmetrical about the optical axis. In the otherpoints, the shape of this lens is similar to that of the firstembodiment.

Thus, the fθ lens of the present invention is wide in the degree offreedom of design with regard to the shape in the sub scanningdirection.

FIG. 7 shows a third embodiment of the fθ lens according to the presentinvention. FIG. 7 shows a cross-sectional view of the lens as it is seenin the main scanning plane. The design values of this lens are shown inTable 5, and the performance of this lens is shown in Table 6.

This embodiment is one in which the thickness t of the lens is madethick as compared with the first embodiment. In the extent of thisembodiment, the influence of environmental fluctuations or the likeposes no problem in practical use. FIG. 8 shows a fourth embodiment ofthe fθ lens according to the present invention. FIG. 8 shows across-sectional view of the lens as it is seen in the main scanningplane. The design values of this lens are shown in Table 7, and theperformance of this lens is shown in Table 8.

This embodiment is one in which the lens is made further thick ascompared with the third embodiment. In this case, t/L is at the utmostpossible limit of the range of expression (2), while t/g is sufficientlywithin the range of expression (3) and therefore, the influence ofenvironmental fluctuations becomes great, but is within a practicalrange.

FIG. 9 shows a fifth embodiment of the fθ lens according to the presentinvention. FIG. 9 shows a cross-sectional view of the lens as it is seenin the main scanning plane. The design values of this lens are shown inTable 9, and the performance of this lens is shown in Table 10.

This lens is such that the incident light is parallel light in the mainscanning plane, and in the sub-scanning plane, it enters so as to beimaged on the deflecting surface of the optical deflector. Therefore,the value of p is greater than in the other embodiments having the samemaximum scanning angle and the same effective scanning width, but thisembodiment has the advantage that aberrations will not extremely changeeven if the incident light beam deviates in the main scanning plane.

FIG. 10 shows a sixth embodiment of the fθ lens according to the presentinvention. FIG. 10 shows a cross-sectional view of the lens as it isseen in the main scanning plane The design values of this lens are shownin Table 11, and the performance of this lens is shown in Table 12.

This embodiment is one in which the maximum scanning angle Qmax isdesigned at 32°. In the case of this lens, r₁ ≧Ymax and therefore, notS₁ (r₁)/r₁ <0.5 of expression (2a) but S₁ (Ymax)/Ymax<0.5×Ymax/r₁ ofexpression (2b) has a meaning. Accordingly, the values of S₁ (Ymax) and0.5×Ymax/r₁ are shown in Table 11.

FIG. 11 shows a seventh embodiment of the fθ lens according to thepresent invention. FIG. 11 shows a cross-sectional view of the lens asit is seen in the main scanning plane. The design values of this lensare shown in Table 13, and the performance of this lens is shown inTable 14.

This embodiment is one in which the maximum scanning angle Qmax isdesigned at 56°.

FIG. 12 shows an eighth embodiment of the fθ lens according to thepresent invention. FIG. 12 shows a cross-sectional view of the lens asit is seen in the main scanning plane. The design values of this lensare shown in Table 15, and the performance of this lens is shown inTable 16.

This embodiment is one in which design is made such that a becomes greatas compared with the other embodiments having the same maximum scanningangle.

The tables of the design values of the embodiments of the fθ lens of thepresent invention will be shown below.

                                      TABLE 1                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        31.64816 mm                                                                           r.sub.1                                                                          26.9119 mm                                                                            r.sub.2                                                                          28.8687 mm                                     t        6.17434 mm                                                                            k.sub.1                                                                          -4.78688                                                                              k.sub.2                                                                          -5.79781                                       b        108.78799 mm                                                                          B.sub.1                                                                          -2.55176 E-6                                                                          B.sub.2                                                                          -3.14086 E-6                                   g        146.61049 mm                                                                          C.sub.1                                                                          -2.88092 E-10                                                                         C.sub.2                                                                          5.02452 E-11                                   θ.sub.max                                                                        44°                                                                            D.sub.1                                                                          1.16473 E-13                                                                          D.sub.2                                                                          1.78545 E-14                                   Y.sub.max                                                                              40.5 mm E.sub.1                                                                          -2.66441 E-18                                                                         E.sub.2                                                                          -3.79296 E-17                                  f.sub.m  321.11485 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        292.79943 mm                                                                          rs ∞ mm                                                                            rs -15.4070 mm                                    Incident light                                                                         Imaged at                                                                             A.sub. 1 '                                                                       0       A.sub.2 '                                                                        1.95908 E-3                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.26105 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -3.47043 E-6                                   t/L      1.02858 C.sub.1 '                                                                        0       C.sub.2 '                                                                        3.04609 E-9                                    t/g      0.04211 D.sub.1 '                                                                        0       D.sub.2 '                                                                        -8.74076 E-13                                  |f.sub.m |/g                                                         2.19026 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -4.84512 E-17                                  p/g      1.99712                                                              __________________________________________________________________________

                                      TABLE 2                                     __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm 0.42491 mm                                                                               0.08531 mm                                                                                  0%                                     11°                                                                         26.82169 mm                                                                            0.18730 mm                                                                              -0.49197 mm                                                                             -0.66041%                                   22°                                                                         53.93793 mm                                                                            0.06700 mm                                                                               0.29648 mm                                                                             -0.11494%                                   33°                                                                         80.91486 mm                                                                            -0.03744 mm                                                                             -0.32326 mm                                                                             -0.10511%                                   44°                                                                         108.09429 mm                                                                           -0.11663 mm                                                                             -1.19237 mm                                                                              0.08731%                                   __________________________________________________________________________

                                      TABLE 3                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        31.64816 mm                                                                           r.sub.1                                                                          26.9119 mm                                                                            r.sub.2                                                                          28.8687 mm                                     t        6.17434 mm                                                                            k.sub.1                                                                          -4.78688                                                                              k.sub.2                                                                          -5.79781                                       b        108.78799 mm                                                                          B.sub.1                                                                          -2.55176 E-6                                                                          B.sub.2                                                                          -3.14086 E-6                                   g        146.61049 mm                                                                          C.sub.1                                                                          -2.88092 E-10                                                                         C.sub.2                                                                          5.02452 E-11                                   θ.sub.max                                                                        44°                                                                            D.sub.1                                                                          1.16473 E-13                                                                          D.sub.2                                                                          1.78545 E-14                                   Y.sub.max                                                                              40.5 mm E.sub.1                                                                          -2.66441 E-18                                                                         E.sub.2                                                                          -3.79296 E-17                                  f.sub.m  321.11485 mm                                                                          --         Sub-scanning side                                                             S.sub.2 ' (y)                                     p        292.79943 mm                                                                          -- --      rs -30.1915 mm                                    Incident light                                                                         Imaged at                                                                             -- --      A.sub.2 '                                                                        9.40209 E-4                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.26105 -- --      B.sub.2 '                                                                        -2.78179 E-6                                   t/L      0.02858 -- --      C.sub.2 '                                                                        2.97918 E-9                                    t/g      0.04211 -- --      D.sub.2 '                                                                        -1.58547 E-12                                  |f.sub.m |/g                                                         2.19026 -- --      E.sub.2 '                                                                        3.34559 E-16                                   p/g      1.99712                                                              __________________________________________________________________________

                                      TABLE 4                                     __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm 0.42491 mm                                                                              0.07271 mm                                                                                   0%                                     11°                                                                         26.82169 mm                                                                            0.18730 mm                                                                              -0.12918 mm                                                                             -0.66041%                                   22°                                                                         53.93793 mm                                                                            0.06700 mm                                                                              0.05200 mm                                                                              -0.11494%                                   33°                                                                         80.91486 mm                                                                            -0.03744 mm                                                                             -1.09368 mm                                                                             -0.10511%                                   44°                                                                         108.09429 mm                                                                           -0.11663 mm                                                                             0.00991 mm                                                                               0.08731%                                   __________________________________________________________________________

                                      TABLE 5                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        30.34728 mm                                                                           r.sub.1                                                                          27.9359 mm                                                                            r.sub.2                                                                          28.2013 mm                                     t        10.00000 mm                                                                           k.sub.1                                                                          -5.12297                                                                              k.sub.2                                                                          -5.57735                                       b        102.72251 mm                                                                          B.sub.1                                                                          -2.62615 E-6                                                                          B.sub.2                                                                          -3.05805 E-6                                   g        143.06979 mm                                                                          C.sub.1                                                                          -2.84538 E-10                                                                         C.sub.2                                                                          2.67583 E-10                                   θ.sub.max                                                                        44°                                                                            D.sub.1                                                                          2.26473 E-13                                                                          D.sub.2                                                                          -2.32333 E-14                                  Y.sub.max                                                                              39.2 mm E.sub.1                                                                          -2.66441 E-18                                                                         E.sub.2                                                                          -2.19162 E-17                                  f.sub.m  350.18611 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        292.79943 mm                                                                          rs ∞ mm                                                                            rs -15.5395 mm                                    Incident light                                                                         Imaged at                                                                             A.sub.1 '                                                                        0       A.sub.2 '                                                                        2.07171 E-3                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.24763 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -3.79789 E-6                                   t/L      0.02315 C.sub.1 '                                                                        0       C.sub.2 '                                                                        3.28469 E-9                                    t/g      0.03495 D.sub.1 '                                                                        0       D.sub.2 '                                                                        -6.34188 E-13                                  |f.sub.m |/g                                                         2.44766 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -2.01964 E-17                                  p/g      2.04655                                                              __________________________________________________________________________

                                      TABLE 6                                     __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm 0.00681 mm                                                                               0.03452 mm                                                                                  0%                                     11°                                                                         26.88949 mm                                                                            -0.11213 mm                                                                             -0.20048 mm                                                                             -0.40930%                                   22°                                                                         54.08625 mm                                                                            0.02424 mm                                                                               0.10438 mm                                                                              0.15972%                                   33°                                                                         80.97892 mm                                                                            0.05452 mm                                                                              -0.25814 mm                                                                             -0.02602%                                   44°                                                                         107.99877 mm                                                                           0.27536 mm                                                                              -1.15897 mm                                                                             -0.00114%                                   __________________________________________________________________________

                                      TABLE 7                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        29.44288 mm                                                                           r.sub.1                                                                          2.95055 mm                                                                            r.sub.2                                                                          27.6168 mm                                     t        15.00000 mm                                                                           k.sub.1                                                                          -5.53373                                                                              k.sub.2                                                                          -5.23709                                       b        94.50704 mm                                                                           B.sub.1                                                                          -2.70055 E-6                                                                          B.sub.2                                                                          -2.83943 E-6                                   g        138.94992 mm                                                                          C.sub.1                                                                          -1.85714 E-10                                                                         C.sub.2                                                                          4.25077 E-10                                   θ.sub.max                                                                        44°                                                                            D.sub.1                                                                          3.09180 E-13                                                                          D.sub.2                                                                          -1.30815 E-14                                  Y.sub.max                                                                              38.3 mm E.sub.1                                                                          -2.42201 E-17                                                                         E.sub.2                                                                          9.11235 E-18                                   f.sub.m  395.46596 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        292.79943 mm                                                                          rs ∞ mm                                                                            rs -15.8563 mm                                    Incident light                                                                         Imaged at                                                                             A.sub. 1 '                                                                       0       A.sub.2 '                                                                        2.12736 E-3                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.23120 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -3.85147 E-6                                   t/L      0.06944 C.sub.1 '                                                                        0       C.sub.2 '                                                                        3.28896 E-9                                    t/g      0.10795 D.sub.1 '                                                                        0       D.sub.2 '                                                                        -6.15462 E-13                                  |f.sub.m |/g                                                         2.84610 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -1.78469 E-16                                  p/g      2.10723                                                              __________________________________________________________________________

                                      TABLE 8                                     __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm 0.01094 mm                                                                               0.00454 mm                                                                                  0%                                     11°                                                                         26.86572 mm                                                                            -0.08744 mm                                                                             -0.02620 mm                                                                             -0.49733%                                   22°                                                                         54.10123 mm                                                                            0.06297 mm                                                                               0.01514 mm                                                                             -0.18746%                                   33°                                                                         80.97640 mm                                                                            0.03416 mm                                                                              -0.02887 mm                                                                             -0.02914%                                   44°                                                                         108.00456 mm                                                                           0.32891 mm                                                                              -0.11629 mm                                                                              0.00422%                                   __________________________________________________________________________

                                      TABLE 9                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        19.61717 mm                                                                           r.sub.1                                                                          22.0311 mm                                                                            r.sub.2                                                                          27.9182 mm                                     t        5.00000 mm                                                                            k.sub.1                                                                          -6.70387                                                                              k.sub.2                                                                          -10.4291                                       b        128.49907 mm                                                                          B.sub.1                                                                          -3.29505 E-7                                                                          B.sub.2                                                                          -2.29333 E-6                                   g        153.11624 mm                                                                          C.sub.1                                                                          -2.46699 E-9                                                                          C.sub.2                                                                          5.95537 E-10                                   θ.sub.max                                                                        44°                                                                            D.sub.1                                                                          1.46251 E-13                                                                          D.sub.2                                                                          6.16143 E-13                                   Y.sub.max                                                                              27.9 mm E.sub.1                                                                          1.37547 E-17                                                                          E.sub.2                                                                          -7.14808 E-16                                  f.sub.m  140.10521 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        ∞ mm                                                                            rs ∞ mm                                                                            rs -11.1244 mm                                    Incident light                                                                         Imaged at                                                                             A.sub. 1 '                                                                       0       A.sub.2 '                                                                        3.59219 E-3                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.26972 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -1.08695 E-5                                   t/L      0.02315 C.sub.1 '                                                                        0       C.sub.2 '                                                                        2.09015 E-8                                    t/g      0.03265 D.sub.1 '                                                                        0       D.sub.2 '                                                                        -4.08754 E-13                                  |f.sub.m |/g                                                         0.91503 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -1.98562 E-14                                  p/g      ∞                                                              __________________________________________________________________________

                                      TABLE 10                                    __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm  0.00188 mm                                                                             -0.03561 mm                                                                                  0%                                     11°                                                                         26.92541 mm                                                                            -0.02322 mm                                                                             0.01809 mm                                                                              -0.27626%                                   22°                                                                         54.06152 mm                                                                             0.01473 mm                                                                             0.17296 mm                                                                               0.11393%                                   33°                                                                         80.98224 mm                                                                            -0.00237 mm                                                                             0.43728 mm                                                                              -0.02193%                                   44°                                                                         107.99953 mm                                                                           -0.02902 mm                                                                             1.03899 mm                                                                              -0.00044%                                   __________________________________________________________________________

                                      TABLE 5                                     __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        71.08404 mm                                                                           r.sub.1                                                                          60.0000 mm                                                                            r.sub.2                                                                          73.3174 mm                                     t        5.00000 mm                                                                            k.sub.1                                                                          -3.87781                                                                              k.sub.2                                                                          -4.34297                                       b        136.16557 mm                                                                          B.sub.1                                                                          -1.84687 E-7                                                                          B.sub.2                                                                          -6.74641 E-7                                   g        212.24962 mm                                                                          C.sub.1                                                                          -8.95896 E-10                                                                         C.sub.2                                                                          -7.69111 E-10                                  θ.sub.max                                                                        32°                                                                            D.sub.1                                                                          2.06800 E-13                                                                          D.sub.2                                                                          2.34907 E-13                                   Y.sub.max                                                                              48.9 mm E.sub.1                                                                          -3.01247 E-17                                                                         E.sub.2                                                                          -5.42806 E-17                                  f.sub.m  505.53622 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        274.92354 mm                                                                          rs ∞ mm                                                                            rs -27.6158 mm                                    Incident light                                                                         Imaged at                                                                             A.sub.1 '                                                                        0       A.sub.2 '                                                                        3.43802 E-4                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (Y.sub.max)/Y.sub.max                                                          0.23093 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -1.61819 E-7                                   0.5 × Y.sub.max /r.sub.1                                                         0.40750 C.sub.1 '                                                                        0       C.sub.2 '                                                                        -3.50099 E-11                                  t/L      0.02315 D.sub.1 '                                                                        0       D.sub.2 '                                                                        3.10358 E-14                                   t/g      0.02356 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -3.04885 E-18                                  |f.sub.m |/g                                                         2.38180                                                              p/g      1.29232                                                              __________________________________________________________________________

                                      TABLE 12                                    __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm 0.00464 mm                                                                               0.01597 mm                                                                                 0%                                       8°                                                                         26.92064 mm                                                                            0.03678 mm                                                                              -0.02679 mm                                                                             -0.29393%                                   16°                                                                         53.93273 mm                                                                            -0.13788 mm                                                                             -0.05082 mm                                                                             -0.12457%                                   24°                                                                         81.05539 mm                                                                            -0.14220 mm                                                                             -0.52391 mm                                                                             0.06738%                                    32°                                                                         108.00432 mm                                                                           0.27305 mm                                                                              -1.18613 mm                                                                             0.00400%                                    __________________________________________________________________________

                                      TABLE 13                                    __________________________________________________________________________    Wavelength       Coefficients of development expression                       used     780 nm  of lens surface                                              Refractive                                                                             1.57467 Lens surface on the                                                                      Lens surface on the                               index of lens    deflection point side                                                                    side of the plane to be                                                       scanned                                           L        216 mm  Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                       a        15.05054 mm                                                                           r.sub.1                                                                          8.94552 mm                                                                            r.sub.2                                                                          7.71858 mm                                     t        5.00000 mm                                                                            k.sub.1                                                                          -7.63139                                                                              k.sub.2                                                                          -7.68973                                       b        86.93520 mm                                                                           B.sub.1                                                                          -9.78039 E-7                                                                          B.sub.2                                                                          -4.97108 E-6                                   g        106.98574 mm                                                                          C.sub.1                                                                          -3.09809 E-9                                                                          C.sub.2                                                                          2.73858 E-9                                    θ.sub.max                                                                        56°                                                                            D.sub.1                                                                          1.79254 E-12                                                                          D.sub.2                                                                          -1.17732 E-12                                  Y.sub.max                                                                              36.4 mm E.sub.1                                                                          -4.3266 E-16                                                                          E.sub.2                                                                          -4.13303 E-17                                  f.sub.m  200.99035 mm                                                                          Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                     p        319.47862 mm                                                                          rs 25.0783 mm                                                                            rs -11.5520 mm                                    Incident light                                                                         Imaged at                                                                             A.sub. 1 '                                                                       0       A.sub.2 '                                                                        1.59426 E-2                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.26884 B.sub.1 '                                                                        0       B.sub.2 '                                                                        -7.46233 E-5                                   t/L      0.02315 C.sub.1 '                                                                        0       C.sub.2 '                                                                        1.45733 E-7                                    t/g      0.04674 D.sub.1 '                                                                        0       D.sub.2 '                                                                        3.30509 E-11                                   |f.sub.m |/g                                                         1.87866 E.sub.1 '                                                                        0       E.sub.2 '                                                                        -7.56819 E-14                                  p/g      2.98618                                                              __________________________________________________________________________

                                      TABLE 14                                    __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm -0.00296 mm                                                                             -0.01150 mm                                                                                 0%                                      14°                                                                         26.96683 mm                                                                            -0.00101 mm                                                                             0.03840 mm                                                                              -0.12285%                                   28°                                                                         54.01959 mm                                                                            -0.00210 mm                                                                             0.08025 mm                                                                              0.03628%                                    42°                                                                         80.99798 mm                                                                            -0.00235 mm                                                                             1.98343 mm                                                                              -0.00249%                                   56°                                                                         108.00401 mm                                                                            0.03719 mm                                                                             -0.12910 mm                                                                             0.00371%                                    __________________________________________________________________________

                                      TABLE 15                                    __________________________________________________________________________    Wavelength        Coefficients of development expression                      used     780 nm   of lens surface                                             Refractive                                                                             1.57467  Lens surface on the                                                                      Lens surface on the                              index of lens     deflection point side                                                                    side of the plane to be                                                       scanned                                          L        216 mm   Main scanning side                                                                       Main scanning side                                                 S.sub.1 (y)                                                                              S.sub.2 (y)                                      a        56.31337 mm                                                                            r.sub.1                                                                          17.1466 mm                                                                            r.sub.2                                                                          14.9427 mm                                    t        5.00000 mm                                                                             k.sub.1                                                                          -3.08642                                                                              k.sub.2                                                                          -2.96849                                      b        72.65824 mm                                                                            B.sub.1                                                                          -1.06905 E-6                                                                          B.sub.2                                                                          -1.19000 E-6                                  g        133.97362 mm                                                                           C.sub.1                                                                          -1.11624 E-9                                                                          C.sub.2                                                                          -8.41386 E-10                                 θ.sub.max                                                                        44°                                                                             D.sub.1                                                                          3.67839 E-13                                                                          D.sub.2                                                                          2.40892 E-13                                  Y.sub.max                                                                              70.4 mm  E.sub.1                                                                          -2.83506 E-17                                                                         E.sub.2                                                                          -1.43905 E-17                                 f.sub.m  -1175.861993 mm                                                                        Sub-scanning side                                                                        Sub-scanning side                                                  S.sub.1 ' (y)                                                                            S.sub.2 ' (y)                                    p        145.00000 mm                                                                           rs ∞ mm                                                                            rs -18.9369 mm                                   Incident light                                                                         Imaged at                                                                              A.sub.1 '                                                                        0       A.sub.2 '                                                                        1.8769 E-3                                    sub-scanning side                                                                      deflection point                                                     S.sub.1 (r.sub.1)/r.sub.1                                                              0.35574  B.sub.1 '                                                                        0       B.sub.2 '                                                                        -2.21157 E-6                                  t/L      0.02315  C.sub.1 '                                                                        0       C.sub.2 '                                                                        6.54907 E-10                                  t/g      0.03732  D.sub.1 '                                                                        0       D.sub.2 '                                                                        2.70727 E-14                                  |f.sub.m |/g                                                         8.77682  E.sub.1 '                                                                        0       E.sub.2 '                                                                        -1.74730 E-17                                 p/g      1.08230                                                              __________________________________________________________________________

                                      TABLE 16                                    __________________________________________________________________________    Incidence     Curvature of image                                                                      Curvature of image                                    angle of      field on the main                                                                       field on the sub-                                                                       Constant speed                              view Imaging position                                                                       scanning plane side                                                                     scanning plane side                                                                     property                                    __________________________________________________________________________     0°                                                                             0 mm  0.02100 mm                                                                             1.21237 mm                                                                                   0%                                     11°                                                                         26.44729 mm                                                                            -0.06439 mm                                                                             1.09819 mm                                                                              -2.04707%                                   22°                                                                         53.92038 mm                                                                            -1.50412 mm                                                                             0.83558 mm                                                                              -0.14744%                                   33°                                                                         81.15944 mm                                                                             0.47430 mm                                                                             0.58732 mm                                                                               0.19684%                                   44°                                                                         107.99872 mm                                                                           -0.35419 mm                                                                             0.55777 mm                                                                              -0.00119%                                   __________________________________________________________________________

FIGS. 13 to 20 show aberrations in the embodiments of the fθ lens of thepresent invention.

Description will now be made of an image forming apparatus using anoptical scanning device to form an image on an image carrier.

FIG. 21 is a cross-sectional view of a laser beam printer which is animage forming apparatus for recording images by a laser beam to whichthe fθ lens of the present invention is applied.

Description will hereinafter be made of FIG. 21.

In FIG. 21, the reference numeral 39 designates a photosensitive drumrotated in the direction of arrow. Around the photosensitive drum 39,there are provided in succession a charger 40 for uniformly charging thesurface of the drum, a developing device 41 for developing a latentimage on the photosensitive drum, a transfer charger 44 for transferringthe image developed by the developing device 41 to paper, and a cleaner46 for removing any unnecessary developer remaining on the surface ofthe photosensitive drum after the image transfer.

The reference numeral 42 denotes a paper supply cassette containingrecording paper therein, the reference numeral 43 designates registerrollers, and the reference numeral 45 denotes a fixating device formaking the transferred image semipermanent. The reference numeral 49designates a fan for forming an air stream on the underside of a housingto thereby prevent the temperature rise of an optical system. Asemiconductor laser output portion 6 contained in a containing portionformed integrally with a base plate 5 outputs a laser beam opticallymodulated by an extraneous signal in conformity with recordinginformation, and this laser beam emerges from a housing comprised of thebase plate 5 and a cover member 27 through a cylindrical lens 33 highlyaccurately held and positioned by a supporting portion formed integrallywith the base plate 5 and the containing portion, a rotatable polygonmirror 2 for deflecting the light beam at a uniform angular velocity,and an fθ lens 31, and scans on the photosensitive drum 39 via a mirror48, whereby a charge pattern conforming to the recording information isformed on the photosensitive drum 39.

The reference numeral 34 denotes a positioning member for the housingprovided on the underside of the base plate 5 integrally with the baseplate, and positioned highly accurately with the photosensitive drum 39and the mirror 48. Positioning accuracy is enhanced by this positioningmember, whereby the positional accuracy relative to the photosensitivedrum is enhanced and therefore, it becomes possible to obtain moreexcellent images.

The charge pattern on the drum is visualized by the developing device41. This visualized image is transferred by the transfer charger 44 ontothe recording paper fed from the paper supply cassette 42, whereafterthe recording paper is fed to the fixating device 45 for fixation of theimage. Any developer remaining on the photosensitive drum after theimage transfer is removed by the cleaner 46 and prepared for anotheruse.

Such an image forming apparatus to which the fθ lens of the presentinvention is applied can be made compact and thin.

As described above, according to the present invention, there can berealized an fθ lens comprising a single lens in which the shape of thecurved surface of at least the lens surface on the deflection point sidein the main scanning plane is an aspherical shape and particularly nearthe optical axis, this aspherical shape becomes convex toward thedeflection point side and further, the shape of this surface in the mainscanning plane is made to satisfy expressions (1) and (2-a) or (2-b),whereby the fθ lens satisfies the performance required of an fθ lens andyet the thickness of the lens is kept small so as to be suitable forplasticization. Also, the use of such fθ lens can realize an imageforming apparatus which can be made compact and thin.

I claim:
 1. An fθ lens acting to effect uniform speed scanning whilecausing a light beam deflected at a uniform angular velocity by thedeflecting surface of a deflector to be imaged on a plane to be scanned,wherein the shape of the curved surface of at least the lens surface ofthe lens which is adjacent to the deflection point side of thedeflecting surface in the main scanning plane is an aspherical shape andparticularly, near the optical axis, in at least the main scanningplane, this aspherical shape is convex toward said deflection point, andwhen the radius of curvature of this convex shape near the optical axisin the main scanning plane is r₁ and the focal length of the fθ lensnear the optical axis in the main scanning plane is fm,

    0≧r.sub.1 <|fm|

and yet, when with the point of intersection between the lens surfaceadjacent to the deflection point side and the optical axis as the originand with the coordinates system of the x-axis plotted in the directionof the optical axis and the coordinates system of the y-axis plotted inthe main scanning plane perpendicularly thereto, the surface shape inthe main scanning plane is expressed as a function S₁ (y) in which y isa variable, and when the maximum effective diameter of this surface inthe main scanning plane is Ymax, S₁ (y) is defined between 0 and Ymax,and when r₁ <Ymax,

    -1<S.sub.1 (r.sub.1)/r.sub.1 <0.5

and when r₁ 24 Ymax,

    -1×Ymax/r.sub.1 <S.sub.1 (Ymax)/Ymax<0.5×Ymax/r.sub.1.


2. An fθ lens according to claim 1, wherein when the thickness of the fθlens on the optical axis thereof is t and the effective scanning widthon the plane to be scanned is L,

    0<t/L<0.08.


3. An fθ lens according to claim 1, wherein when the thickness of the fθlens on the optical axis thereof is t and the distance from thedeflection point of the deflecting surface to the plane to be scanned isg,

    0<t/g<0.15


4. An fθ lens according to claim 1, wherein the focal length fm near theoptical axis in the main scanning plane and the distance g from thedeflection point of the deflecting surface to the plane to be scannedare

    |fm|/g>0.8


5. An fθ lens according to claim 1, wherein the shape of the lens nearthe optical axis is a meniscus lens convex toward the deflection pointside of the deflecting surface in at least the main scanning plane. 6.An fθ lens according to claim 1, wherein the light beam incident on thelens is substantially convergent light in at least the main scanningplane.
 7. An fθ lens according to claim 6, wherein the light beamincident on the lens is substantially convergent light in at least themain scanning plane, and the distance p from the deflection point of thedeflecting surface to a point at which the convergent light is imaged inthe main scanning plane and the distance g from the deflection point ofthe deflecting surface to the plane to be scanned are

    0.7<p/g.


8. An fθ lens according to claim 1, wherein the light beam incident onthe lens is substantially parallel light in at least the main scanningplane.
 9. An fθ lens according to claim 1, wherein with regard to atleast one of a curved surface adjacent to the deflection point and acurved surface adjacent to the plane to be scanned, the radius ofcurvature near the optical axis differs between the main scanningcross-section and the sub-scanning cross-section (a plane containing theoptical axis of the lens and perpendicular to the main scanning plane).10. An fθ lens according to claim 9, wherein with regard to at least oneof a curved surface adjacent to the deflection point and a curvedsurface adjacent to the plane to be scanned, the radius of curvature ofthe curved surface of the lens determined in a plane containing a normalin the main scanning plane and perpendicular to the main scanning planevaries continuously in the effective portion of the lens.
 11. An fθ lensaccording to claim 9 or 10, wherein in the sub-scanning cross-sectionnear the optical axis, the deflection point and the plane to be scannedare in an optically substantially conjugate relation.
 12. A scanningoptical system comprising a light source, a first optical system forcondensing light from the light source and causing the light to emergetoward the deflection point of the deflecting surface of a deflector, adeflector for deflecting the light beam from the first optical system ata uniform angular velocity on the deflection point of the deflectingsurface thereof, a second optical system comprising an fθ lens acting toeffect uniform speed scanning while causing the light beam deflected ata uniform angular velocity from the deflection point of the deflector tobe imaged on a plane to be scanned, and a plane to be scanned on which arecording material such as a photosensitive member is placed,characterized in that the shape of the curved surface in the mainscanning plane of at least the lens surface of the fθ lens of the secondoptical system which is adjacent to the deflection point is anaspherical shape and particularly near the optical axis, in at least themain scanning plane, this aspherical shape is convex toward thedeflection point, and when the radius of curvature of this convex shapenear the optical axis in the main scanning plane is r₁ and the focallength of said fθ lens near the optical axis in the main scanning planeis fm,

    0≦r.sub.1 <|fm|

and yet, when with the point of intersection between the lens surfaceadjacent to said deflection point side and the optical axis as theorigin, the coordinates system of the x-axis is plotted in the directionof the optical axis and the coordinates system of the y-axis is plottedin the main scanning plane perpendicularly thereto and when the surfaceshape in the main scanning plane is expressed as a function S₁ (y) inwhich y is a variable and when the maximum effective diameter of saidsurface in the main scanning plane is Ymax, S₁ (y) is defined between 0and 5 Ymax, and when r₁ <Ymax,

    -1<S.sub.1 (r.sub.1)/r.sub.1 <0.5

and when r₁ >Ymax,

    -1×Ymax/r.sub.1 <S.sub.1 (Ymax)/Ymax<0.5×Ymax/r.sub.1.


13. A scanning optical system according to claim 12, wherein when thethickness of the fθ lens of the second optical system on the optic axisthereof is t and the effective scanning width on the plane to be scannedis L,

    0<t/L<0.08.


14. A scanning optical system according to claim 12, wherein when thethickness of the fθ lens of the second optical system on the opticalaxis thereof is t and the distance from the deflection point to theplane to be scanned is g,

    0<t/g<0.15.


15. A scanning optical system according to claim 12, wherein the focallength fm of the fθ lens of the second optical system near the opticalaxis in the main scanning plane and the distance g from the deflectionpoint to the plane to be scanned are

    |fm|/g>0.8.


16. A scanning optical system according to claim 12, wherein the fθ lensof the second optical system is an fθ lens in which the shape of thelens near the optical axis is a meniscus lens convex toward thedeflection point side in at least the main scanning plane.
 17. Ascanning optical system according to claim 12, wherein the light beamemerging from the first optical system is substantially convergent lightin at least the main scanning plane.
 18. A scanning optical systemaccording to claim 17, wherein the light beam emerging from the firstoptical system is substantially convergent light in at least the mainscanning plane, and the distance p from the deflection point to a pointat which said convergent light is imaged in the main scanning plane andthe distance g from the deflection point to the plane to be scanned are

    0.7<p/g.


19. A scanning optical system according to claim 12, wherein the lightbeam emerging from the first optical system is substantially parallellight in at least the main scanning plane.
 20. A scanning optical systemaccording to claim 12, wherein the fθ lens of the second optical systemis an fθ lens in which with regard to at least one of a curved surfaceadjacent to the deflection point and a curved surface adjacent to theplane to be scanned, the radius of curvature near the optical axisdiffers between the main scanning cross-section and the sub-scanningcross-section.
 21. A scanning optical system according to claim 20,wherein the fθ lens of the second optical system is such that withregard to at least one of the curved surface adjacent to the deflectionpoint and the curved surface adjacent to the plane to be scanned, theradius of curvature of the curved surface of the lens determined in aplane containing a normal in the main scanning plane and perpendicularto the main scanning plane varies continuously in the effective portionof the lens.
 22. A scanning optical system according to claim 20 or 21,wherein the light beam emerging from the first optical system iscondensed substantially on the deflection point of the deflector in thesub-scanning cross-section and yet, the fθ lens of the second opticalsystem is such that in the sub-scanning cross-section near the opticalaxis, the deflection point and the plane to be scanned are in anoptically substantially conjugate relation.
 23. A scanning opticalsystem according to any one of claims 12 to 22, wherein a semiconductorlaser is used as the light source.
 24. A scanning optical systemaccording to claim 12, incorporated as an input means to an imageforming apparatus that effects an image formation.